It is the unique element generating the characteristic ideal of \(X(E/\mathbb {Q}^{cyc})\)  which is a power of \(p\)  times a distinguished polynomial. It is expected that when \(E\)  is an elliptic curve for which \(E[p]\)  is irreducible, then \(\mu _{E}^{alg}=0\) . It is well known that the condition \(\mu _{E}^{alg}=0\)  depends only on the residual representation \(E[p]\)  (see [1]}). That is, if two elliptic curves \(E_1\)  and \(E_2\)  over \(\mathbb {Q}\)  with good ordinary reduction at \(p\)  have isomorphic residual representation at \(p\) , then \(\mu _{E_1}^{alg}=0\)  if and only if \(\mu _{E_2}^{alg}=0\) . The \(\lambda \) -invariant \(\lambda _{E}^{alg}\)  need not be zero and moreover, need not depend on the residual representation (see [1]}). We define Iwasawa invariants \(\mu _{E,\Sigma _0}^{alg}\)  and \(\lambda _{E,\Sigma _0}^{alg}\)  associated to the \(\Sigma _0\) -imprimitive dual Selmer group \(X^{\Sigma _0}(E/\mathbb {Q}^{cyc})\) . For \(l\in \Sigma _0\) , let \(h_E^{(l)}(T)\)  be the characteristic polynomial of the Pontryajin dual of the \(\mathcal {H}_l(\mathbb {Q}^{cyc})\) . Let \(\sigma _E^{(l)}\)  be such that \(\mathcal {H}_l(\mathbb {Q}^{cyc})\simeq (\mathbb {Q}_p/\mathbb {Z}_p)^{\sigma _{E}^{(l)}}\)  as in [1]}. The characteristic polynomial of the \(\Lambda \) -module \(\operatorname{Sel}^{\Sigma _0}(E/\mathbb {Q}^{cyc})\)  is denoted by \(f_{E,\Sigma _0}^{alg}(T)\) . There is a short exact sequence
\(0\rightarrow \operatorname{Sel}(E/\mathbb {Q}^{cyc})\rightarrow \operatorname{Sel}^{\Sigma _0}(E/\mathbb {Q}^{cyc})\rightarrow \prod _{l\in \Sigma _0} \mathcal {H}_l(\mathbb {Q}^{cyc})\rightarrow 0.\) 
